1. If the car is in your name it is safe, providing you have not just transferred ownership. Mind you with a value of £1,000 they would more often than not be interested in it.
2. Might be best. It may also be looked at to see if there are any unusual transactions. It may be worth you opening a new account that has no history?
3. The income and expenditure on the Statement of Affairs asks about other contributions. If he is made BR all his debts will be taken into account. If any are in joint names, the creditors will just target you.